Symmetries of a physical system simplify the task of finding solutions of a theory. While classical physical systems have continuous group symmetries, these symmetries are broken on quantization. A discrete subgroup of the original symmetry group often remains as a symmetry of the quantized theory.
For example, in the low-energy effective limit of M-theory, namely eleven-dimensional supergravity and its compactifications, discrete symmetry groups are conjectured to encode the quantum corrections. The symmetry groups in compactification to dimensions greater than or equal to 3 are discrete subgroups of finite dimensional Lie groups. Furthermore, the quantum corrected supergravity action is expressed in terms of automorphic forms. By continuing dimensional reduction small dimensions, infinite dimensional Kac-Moody group symmetry begins to emerge.
There is strong evidence and a number of concrete conjectures indicating that Kac-Moody groups encode the symmetries of eleven-dimensional supergravity and M-theory. Our research in this direction focusses on developing the mathematical techniques and structures required to investigate these conjectures.
Gravity theories and their symmetries in small dimensions are particularly amenable to explicit computation and investigations. Here we focus on the geometry and symmetry of de Sitter space as a solution of Einstein's equation with a positive cosmological constant.